Let be a twice-differentiable function that is concave up on the closed interval

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let ff be a twice-differentiable function that is concave up on the closed interval [2,6][2,6]. The interval is divided into nn equal subintervals, and both the midpoint and trapezoidal approximations to ∫26f(x) dx\displaystyle \int_{2}^{6} f(x)\,dx are computed. Which of the following correctly describes the relative sizes of these two approximations and the exact value of the integral? (a) Trapezoidal approximation < <\, exact value < <\, midpoint approximation (b) Exact value < <\, midpoint approximation < <\, trapezoidal approximation (c) Midpoint approximation < <\, exact value < <\, trapezoidal approximation (d) Both the midpoint and trapezoidal approximations exceed the exact value

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